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Powers of Lucas numbers.
1

%I #16 Sep 01 2024 20:37:53

%S 1,2,3,4,7,8,9,11,16,18,27,29,32,47,49,64,76,81,121,123,128,199,243,

%T 256,322,324,343,512,521,729,841,843,1024,1331,1364,2048,2187,2207,

%U 2209,2401,3571,4096,5776,5778,5832,6561,8192,9349,14641,15127,15129,16384

%N Powers of Lucas numbers.

%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 56.

%D Thomas Koshy, "Fibonacci and Lucas Numbers and Applications", Wiley, New York, 2001.

%D V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.

%H T. D. Noe, <a href="/A105941/b105941.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LucasNumber.html">Lucas Number.</a>

%F {a(n)} = {A000204} U {A001254} U {A075155} U {A099923} U {A103325}... L(n)^2 = L(2n) + 2(-1)^n = L(n-1)*L(n+1) + 5(-1)^n. L(n)^3 = L(3n) + 3(-1)^n*L(n). L(n)^4 = L(4n) + 4(-1)^n*L(2n) + 6. L(n)^5 = L(5n) + 5(-1)^n*L(3n) + 10L(n).

%t lim = 10^5; t = Table[f = LucasL[n]; If[f == 1, {1}, f^Range[Floor[Log[lim]/Log[f]]]], {n, 0, Floor[Log[GoldenRatio, lim]]}]; Union[Flatten[t]] (* _T. D. Noe_, Sep 27 2011 *)

%Y A000032 Lucas numbers. A001254 Squares of Lucas numbers. A075155 Cubes of Lucas numbers. A099923 Fourth powers of Lucas numbers. A103325 Fifth powers of Lucas numbers. A103324 Square array T(n, k) read by antidiagonals: powers of Lucas numbers. A105317 Powers of Fibonacci numbers.

%Y Cf. A000032, A001254, A075155, A099923, A103325, A103324, A105317.

%K easy,nonn

%O 1,2

%A _Jonathan Vos Post_, Apr 27 2005

%E Corrected by _T. D. Noe_, Sep 26 2011